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Measurement

Measurement. Exponential Notation Decimal coefficient + Power of 10 multiplier Significant Figures Accuracy and precision Rounding Off Rules for rounding up and down Dimensional Analysis Mind the units being converted Accuracy and Precision

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Measurement

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  1. Measurement • Exponential Notation • Decimal coefficient + Power of 10 multiplier • Significant Figures • Accuracy and precision • Rounding Off • Rules for rounding up and down • Dimensional Analysis • Mind the units being converted • Accuracy and Precision • Average, max, min, deviation, standard deviation

  2. Number Notation • Common symbols in text books • 102 = 100, • √25 = 5 • Calculators and computers (e.g. Excel) use other conventional symbols • 100 = 10^2 or 10E2 (Excel) =10exp2 (Casio) • 25^0.5 = 25E0.5 = 25^(1/2) for square roots • yx also does ANY powers & roots

  3. Why use Exponents? • Huge range of values in nature • 299,792,458 meters/sec speed of light • 602,214,200,000,000,000,000,000 atoms/mole • 0.000000625 meters, wavelength of red light • 0.0000000000000000001602 electron charge • Much simpler to utilize powers of 10 • 3.00*108 meters/sec speed of light • 6.02*1023 atoms/mole • 6.25*10-7 meters for wavelength red light • 1.60*10-19 Coulombs for electron’s charge

  4. People like small numbers • Tend to think in 3’s • good, better, best (Sears appliances) • Small, medium, large (T-shirts, coffee serving) • 1-3 digit numbers easier to remember • Temperature, weight, volume • Modifiers turn big back into small numbers • 2000 lb  1 ton, 5280 feet  1 mile • Kilograms, Megabytes, Gigahertz, picoliters (ink jet)

  5. SI metric prefix nomenclature

  6. more SI prefixes (also on Jaguar web)

  7. Exponential or Scientific Notationkeeps numbers relatively simple • Decimal number identifying significant digits • Example: 5,050,520 • Exponent of 10 identifies overall magnitude • Example: 10^6 or E6 (denoting 1 million) • Combined expression gives entire value • 5.05052 x 106 (usual text book notation) • 5.05052*10^6 (computers, Excel) • 5.05052*10exp6 (some calculators) • 5.05052E6 (alternative in Excel)

  8. Exponential Notation Notation method Single digit (typically) before decimal point Significant digits (2-3 typical) after decimal Power of 10 after the significant digits More Examples 1,234 = 1.234 x 103 = 1.234E3 (Excel) 0.0001234 = 1.234 x 10-4 = 1.234E-4 6-7/8 inch hat size, in decimal notation 6+7/8 = 6+0.875 = 6.875 inch decimal equivalent 6.875, also OK is 0.6875E1 = 6.875E0 = 68.75E-1

  9. Exponential Notation 3100 x 210 = 651,000 In Scientific Notation: 3.100E3 x 2.10E2 Coefficients handled as usual numbers 3. 100 x 2.10  6.51 with 3 significant digits Exponents add when values multiplied 1,000 * 100 = 100,000 10 3 * 10 2 = 10 5 E3 (1,000) * E2 (100) = E5 (100,000) Asterisk (*) indicates multiplication in Excel Final answer is 6.51E5 = 6.51*10^5 NO ambiguity of result or accuracy

  10. Exponential Notation Exponents subtract in division E3 (1,000) / E2 (100) = E1 (10) Forward slash (/) indicates division Computers multiply & divide FIRST Example 1+2*3= 7, not 9 Example (1+2)*3 = 9 Work inside parenthesis always done first Use (extra) parenthesis to avoid errors

  11. Measurement • Exponential Notation • Decimal coefficient + Power of 10 multiplier • Significant Figures • Accuracy and precision • Rounding Off • Rules for rounding up and down • Dimensional Analysis • Mind the units

  12. Significant Figures • Precision must be tailored for the situation • Result cannot be more precise than input data • Data has certain + uncertain aspects • Certain digits are known for sure • Final (missing) digit is the uncertain one • 2/3 cups of flour (intent is not 0.66666666667) • Fraction is exact, but unlimited precision not intended • Context says the most certain part is 0.6 • Uncertain part is probably the 2nd digit • Recipe probably works with 0.6 or 0.7 cups • How to get rid of ambiguity?

  13. Significant Figures • “Sig Figs” = establish values of realistic influence • 1cup sugar to 3 flour does not require exact ratio of 0.3333333 • Unintended accuracy termed “superfluous precision” • Need to define actual measurement precision intended • “Cup of flour” in recipe could be +/- 10% or 0.9 to 1.1 cup • Can’t be more Sig-Figs than least accurate measure • Final “Sig Fig” is “Uncertainty Digit” … least accurately known • adding .000001 gram sugar to 1.1 gram flour = 1.1 gram mixture

  14. How to Interpret Sig-Figs(mostly common sense) • All nonzero digits are significant • 1.234 g has 4 significant figures, • 1.2 g has 2 significant figures. • “0” between nonzero digits significant: • 3.07 Liters has 3 significant figures. • 1002 kilograms has 4 significant figures

  15. Handling zeros in Sig-Figs • Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point: • 0.001 oC has only 1 significant figure • 0.012 g has 2 significant figures • 1.51 nanometers (0.00000000151 meter), 3 sig figs • Trailing zeroes that are to theright of a decimal point with numerical values are always significant: • 0.0230 mL has 3 significant figures • 0.20 g has 2 significant figures • 1.510 nanometers (0.000000001510 meters), 4 sig figs

  16. More examples with zeros • Leading zeros don’t count • Often just a scale factor (0.000001 = microgram) • Middle zeros between numbers always count • 1.001 measurement has 4 decades of accuracy • Trailing zeros MIGHT count • YES if part of measured or defined value, 1.001 • YES if placed intentionally, 7000 grains ≡ 1 pound • NO if zeros to right of non-decimal point • 1,000 has 1 sig-fig … but 1,000.0 has 5 sig-figs • NO if only to demonstrate scale • Carl Sagan’s “BILLIONS and BILLIONS of stars” • Does NOT mean “BILLIONS” + 1 = 1,000,000,001

  17. More Sig-Fig ExamplesHow many sig figs below? • Zeros between • 60.8 has __ significant figures • 39008 has __ sig-figs • Zeros in front • 0.093827 has __ sig-figs • 0.0008 has __ sig-fig • 0.012 has __ sig-figs • Zeros at end • 35.00 has __ sig-figs • 8,000.000 has __ sig-figs • 1,000 has ___ sig figs

  18. More Sig-Fig Examples Zeros between 60.8 has 3 significant figures 39008 has 5 sig-figs Zeros in front 0.093827 has 5 sig-figs 0.0008 has 1 sig-fig 0.012 has 2 sig-figs Zeros at end 35.00 has 4 sig-figs 8,000.000 has 7 sig-figs 1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4

  19. Sig-Fig Exponential Notation • A number ending with zeroes NOT to right of decimal point are not necessarily significant: • 190 miles could be 2 or 3 significant figures • 50,600 calories could be 3, 4, or 5 sig-figs • Ambiguity is avoided using exponentialnotation to exactly define significant figures of 3, 4, or 5 by writing 50,600 calories as: • 5.06 × 10E4 calories (3 significant figures) or • 5.060 × 10E4 calories (4 significant figures), or • 5.0600 × 10E4 calories (5 significant figures). • Remember values right of decimal ARE significant

  20. Exact Values • Some numbers are exact because they are known with complete certainty. • Most exact numbers are simple integers: • 12 inches per foot, 12 eggs per dozen, 3 legs to a tripod • Exact numbers are considered to have an infinite number of significant figures. • When using an exact number in a calculation, the idea of significant figures for that item is ignored when determining the number of significant figures in the result of a calculation • 2.54 cm per inch (exact, NOT 3 sig figs) • 5/9 Centigrade/Fahrenheit degree (exact) • 5280 feet per mile (exact, based on definitions) • The challenge is to remember which numbers are exact

  21. more Sig-Fig Accounting • Addition & Subtraction • Least Significant Figure determines outcome • 1.01 + 1.00000001 = 2.01 • Multiplication & Division • Least Significant Figure determines outcome • 1.01 x 1.0000001 = 1.01 • Round-Off • Calculations can yield more sig-figs than justified • Must reduce answer to lowest sig-fig component

  22. Sig-Fig Multiply & Divide Good first step to use scientific notation Multiply 0.113 * 5280  1.13E-1 * 5.280E3 Multiply the leading values, add the exponents Becomes 5.96640E2 (or 596.64) Sig.Fig. set by least precise input  5.96E2 Divide 4995 by .0012  4.995E3 / 1.2E-3 Divide leading values, subtract the exponents Becomes 4.1625E6 (4,162,500 zeros=magnitude) Sig.Fig. set by least precise input  4.2E6

  23. Sig-Fig Addition & Subtraction First get the decimals (blue #) to align Take 1.0234E3 same as 1,023.4 Then add 1.0E-4 same as + 0.0001 Then subtract 15.22 same as - 15.22 Do the math 1,008.1803 Round to least decimal sig fig 1,008.2 “spitting in the ocean” analogy … if you measure ocean volume by cubic meters or miles, adding a teaspoon is undetectable !

  24. Measurement • Exponential Notation • Decimal coefficient + Power of 10 multiplier • Significant Figures • Accuracy and precision • Rounding Off • Rules for rounding up and down • Dimensional Analysis • Mind the units

  25. Partial Values • Averages, fractions, yields • 2/3 cups flour = 0.66666666666666 …cups? • >2 digit precision inappropriate for cookies • See Mrs. Fields Cookie Recipe • “superfluous accuracy” • unjustified or unwarranted level of detail • Precision needs to fit the situation • “Rounding Off” to appropriate accuracy • Need rules to set the values

  26. Mostly single digits in recipe, no tolerances on measurements, so what is intended degree of precision? 1 cup flour could be 0.5 cup (rounded up to 1) to 1.4 cup (rounded down to 1) ½ cup sugar (0.5 cup) could be 0.45 to 0.54 cup per rounding rules 1/3 cup chocolate chips may be exactly 0.3333333333333333 cups, probably not. Context (& common sense) must be used to interpret precision of values seen in daily life. jj

  27. more Sig-Fig Accounting • Round-Off • Calculations can yield more sig-figs than justified • Must reduce result to lowest sig-fig component • Methodology (usual & customary rules) • If value beyond last sig-fig is ≥5, round UP • For 3 sig-fig accuracy, 5.255123 becomes 5.26 • If value beyond last sig-fig is <5, round OFF • For 3 sig-figs accuracy, 5.254459 becomes 5.25

  28. Rounding Rules … Traditional Rule is Simplest • When trailing digit is <5 round down/off • 1.244 rounded to 3 digits  1.24 • 1.2449999 rounded to 3 digits  1.24 • When trailing digit is ≥5 round up • 1.246 rounded to 3 digits  1.25 • 1.2460111 rounded to 3 digits  1.25 • Note lack of symmetry at “5” • 5 is in the middle, but rounds up • Unintended bias is towards larger values

  29. Guidelines for using calculators • Don’t round off too soon, do it at end of calculation • (5.00 / 1.235) + 3.000 + (6.35 / 4.0) • 4.04858 + 3.000 + 1.5875 = 8.630829  8.6 • 1st division results in 3 sig-figs, last division results in 2 sig-figs. • 3 numbers added should result in 1 digit after the decimal. Thus, the correct rounded final result should be 8.6. This final result has been limited by the accuracy in the last division. • Warning: carrying all digits through to the final result before rounding is critical for many mathematical operations in statistics. Rounding intermediate results when calculating sums of squares can seriously compromise the accuracy of the result.

  30. Don’t round off until the endExample shows common error

  31. Measurement • Exponential Notation • Decimal coefficient + Power of 10 multiplier • Significant Figures • Accuracy and precision • Rounding Off • Rules for rounding up and down • Dimensional Analysis • Mind the units

  32. Dimensional Analysis • Making the units come out right • Useful strategy to avoid calculation errors • Relies on “cancellation of dimensions” • If sec^2 instead of sec/sec cancel, something got inverted • Should always put dimensions on initial formulas • Good News • Easy to do • Avoids silly answers with wrong dimensions. • Bad News • Does not insure right physical relationships • No guarantee of right answer … but units OK

  33. Dimensional Analysis Human Body Temperature Accepted healthy value in USA is 98.6oF Convert to Celsius: (98.6 – 32) oF * (5oC/9oF) = 37.0oC Accepted (customary) value in Europe is 37oC Convert to Fahrenheit (37oC * 9oF/5oC) + 32oF = 99oF Result is 2 sig-figs, and an apparent temperature rise What happened… are Europeans hotter? 2 digit sig-fig on a larger unit of measure (oC), vs 3 sig figs on smaller degree (oF) is inconsistent. Europeans might argue that variability between healthy people negates need for higher sig fig.

  34. Dimensional Analysis Speed Limit 100 km/hr vs. miles/hr (e.g. CA auto in Europe, Mexico, or Canada) (100 km/hr *1000 m/km *100 cm/m) / (2.54 cm/inch*12 inch/foot*5280 foot/mile) = 62.13711922 mph If 100 km/hr posted limit exact (e.g. 100.00000 …) An exact value leads to infinite precision 62.13711922 … Mathematically correct, but impractical for speedometers If 100 km/hr posted speed limit is NOT exact 3 sig fig limit sets speed at 62.1 mph 2 sig fig limit sets speed at 62 mph 1 sig fig sets speed limit at 60 mph

  35. Addition & Subtraction (0.0048965 – 0.00347) x (3.248E4 – 4.58983E3) • Solve what’s inside parenthesis FIRST • Initial value 1st parenthesis 0.0048965 4.8965 E-3 • Subtract 2nd value 0.00347 3.47 E-3 • Result after subtraction 0.0014265 1.4265 E-3 • Round to least accurate 0.00143 1.43 E-3 • Second Parenthesis Calculation • 3.248E4 same as 32,480 32.48 E3 • Subtract 4.58983E3 same as 4,589.83 - 4.58983 E3 • Result after subtraction 27,890.17 27.89017 E3 • Round to low of 4 sig fig 27,890 27.89 E3 • Multiply results from parenthesis calculations • 0.00143 * 27,890 = 39.88270  39.9 • Multiplication accuracy limited to least sig figs = 3 in this case

  36. Accuracy and Precision • Accuracy is the degree of conformity of a measured or calculated quantity to its actual (true) value. • Precision, also called reproducibility or repeatability, is the degree to which further measurements or calculations show the same or similar results. • A measurement can be accurate but not precise; precise but not accurate; neither; or both. • Show water slide video … is he accurate or precise?

  37. Accuracy • Degree of error in achieving the established measurement goal • The Cubit average value has not changed much since biblical times at about 18 inches so it has remained relatively accurate over hundreds (perhaps thousands) of years.

  38. Good accuracy This example shows good accuracy, but low precision

  39. Precision • How well multiple measurements agree with one another to provide a consistent value. (e.g. tight grouping, low dispersion, “all together” series of events). • The “cubit” is not a very precise measure of distance, since it varies between observers using the same definition. No two people are the same, so length data is dispersed. (e.g. inconsistent individual measurements).

  40. Cubits in Class • Historical unit of measure was “cubit” • Distance between elbow and middle finger • Biblical refrerences (Noah’s ark) • Used by pyramid builders in Egypt • What is precision and accuracy? • Let’s take some data !

  41. Reviewing Cubit Data • Our class data is a useful example • Accuracy was good, agrees with history • Historical value 18 inches • Precision not as good, a lot of variation • Size varies greatly between individuals • Average is more consistent

  42. Target analogyThis example has high precision, but poor accuracy

  43. Accuracy versus Precision

  44. Big Slip Video • Is this example precise or accurate? • Experiment repeated by Myth Busters • Performed experiment into a lake • Missing target in water less damaging than hitting dirt if you miss the wading pool! • Show Video ….

  45. Statistical Terms • Mean value • Sum of measurements divided by their number (18.4) • Mode • The most common or “popular” value (18.5 twice) • Range • The largest versus smallest measured value (5.5) • Deviation (from the average) • How each measurement differs from average • Standard Deviation • A mathematical way to minimize influence of “flyers” • Our result was 1.6, a lot less than range of 5.5 • “RMS” (Root Mean Square) used a lot in Engineering

  46. Standard Deviation, why bother? Range a poor indicator of accuracy One bad measurement controls the range Averaging scheme redefines error RMS (root mean squared) is common tool Moves error to an average value basis Suppresses random error contribution

  47. Deviation & Variation Variance is the average of the squared differences between data points and the mean. Variance is tabulated in units squared. Standard deviation is the square root of the sum of variances, and measures the spread of data about the mean, with the same units. Said more formally, the standard deviation is the root mean square (RMS) deviation of values from their arithmetic mean.

  48. Standard Deviation

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